Stability of Tsallis antropy and instabilities of Renyi and normalized Tsallis entropies: A basis for q-exponential distributions

TL;DR

This paper investigates the stability of different generalized entropies and finds that Tsallis entropy uniquely provides a stable basis for q-exponential distributions observed in complex systems.

Contribution

It demonstrates that among the considered entropies, only Tsallis entropy is stable under small distribution deformations, clarifying its suitability for physical applications.

Findings

Tsallis entropy is stable under small perturbations.

Renyi and normalized Tsallis entropies are unstable.

Stability analysis clarifies the physical relevance of Tsallis entropy.

Abstract

The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Renyi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.